THE FORMAL REASONING OF QUANTUM MECHANICS: CAN WE MAKE IT
CONCRETE? SHOULD WE?
Dean Zollman, Kansas State University, USA
The teaching
and learning of quantum mechanics is very frequently postponed until relatively
late in a student’s academic career. In U.S. universities students typically
receive a quick introduction to some aspects of one-dimensional quantum
mechanics from the end of the second or beginning of the third year of their
university studies and then do not study quantum mechanics in any depth until
the fourth year. Thus, the major concepts which have driven much of the
development of physics and of modern technology during the 20thcentury are
delayed until the end of a physicist’s academic career and are frequently not
studied by other students at any time during their careers.
One reason
for this delay is the rather abstract nature of quantum mechanics itself. We
can easily argue that, for the way in which quantum mechanics is traditionally
taught, students need to have generally developed their formal reasoning
skills. For example, formal operations, in the Piagetian sense, include
hypothetical and deductive reasoning, abstract thought, use of symbolic
representation, and the use of transformations. Quantum mechanics is a
hypothetical system for understanding very small objects. It relies heavily on
the use of symbolic representations and deduction to apply quantum mechanics to
a variety of situations. Symmetry arguments, and therefore transformations, are
a significant part of many presentations of quantum mechanics. Therefore
overall, we can assume that the traditional mode in which quantum mechanics is
taught is very abstract and requires rather sophisticated formal operational
procedures.
Significant
research dating back to the 1970s has shown that many university students have
not yet developed formal operations (McKinnon & Renner 1971). In fact, the
traditional way of teaching classical physics is a significant mismatch for
many of these concrete operational students. Thus, it is not surprising that
many physicists conclude that quantum mechanics is not understandable by
students who are not studying physics very carefully and are in their third or
fourth year at a university. Many people have concluded that learning quantum
mechanics at a lower level is not possible and thus should not even be
attempted (Arons 1990). They argue that the students will only be able to
memorize isolated facts and repeat things without true understanding. Thus, the
students are better served if we spend all of our time on classical physics
where concrete learning experiences can more easily be constructed rather than
attempting to teach them something that they could learn only with great
difficulty, if at all.
1. Why teach
quantum mechanics to non-physicists?
The
discussion about the abstract nature of the normal presentations of quantum
mechanics seems rather valid. The simplest response to these conclusions is to
avoid teaching this topic at any but the most advanced levels. However, some
arguments favour attempting to find ways to teach the topic to students who
have not yet reached full formal operations. For example, quantum mechanics was
the most important development in 20thCentury physics, and it has dominated
physics and technology for well over a half a century. Thus, at the beginning
of the 21stCentury it is time to allow all interested people access to these
ideas. Further, many experts predict that within the next 10 years
miniaturization of electronics will reach the quantum mechanics limit. It would
be nice if people who are trying to take the next step – development or
business – understood what that meant. Finally, many other very complex and
abstract processes – the election of an American President, for example – fill
our lives. Perhaps an appreciation of quantum physics can help us understand
the role of measurement in these events.
2. Making
quantum mechanics concrete
Our group at
Kansas State University has been convinced that we should make the teaching of
quantum mechanics more concrete than it normally is. We have worked to develop
both the pedagogical style and the presentation of content so that students who
are still developing their formal operational skills can appreciate and
understand some of the features of contemporary quantum physics. For a
pedagogical strategy we have adopted the basic Learning Cycle. The Learning
Cycle was developed by Robert Karplus about 30 years ago and has been
successfully used in almost all levels of teaching (Karplus 1977, Karplus et al
1975, Zollman 1990). While many people have adapted or changed the basic
Learning Cycle, we find that the one that Karplus originally introduced works
quite well for our teaching situation. Each Learning Cycle begins with an
Exploration where students complete activities prior to the introduction of a
new concept. These activities prepare them for the introduction of new concepts
which can explain their observations during the Exploration. The concept
introduction provides the new principles on which the students will build and
frequently includes the development of models that can help explain the
observations. Once the students have the new concepts and models they complete
an Application in which they apply the newly learned information to situations
that are similar but not identical to the ones they have already studied.
With the
addition of model building in some cases our Learning Cycle comes very close to
the Modelling Cycle that has been developed by Hestenes and his co-workers
(Wells et al 1995). We also emphasize collaboration among students as they are
learning. This cooperative effort is also an important part of the Modelling
Cycle.
We have
created Learning Cycles for a variety of different types of students. Our basic
approach is that all students, even those who are more advanced in their
reasoning skills and academic careers, can profit from a more concrete or
intuitive approach to the abstract ideas of quantum physics. We began by
developing materials for secondary school students and those university
students who would complete a physics course but not study physics beyond one
year. As these materials were developed, they were used by faculty who were
teaching higher-level courses to physics students. We then created a set of
materials for that group and have recently expanded to include materials
specifically aimed at medical students and physics students in the last year of
their undergraduate university careers. Each set of materials has a somewhat
different approach and a different level of mathematical sophistication.
However, all of them follow a basic Learning Cycle and focus on concrete
visualization rather than abstract mathematical deduction.
3. Device
orientation
One way to
make abstract ideas concrete is to connect the concept directly to something in
the students’ experiences. In one way such a connection is easy for quantum
physics. Almost every contemporary technological device could not exist if a
designer of that device did not have an understanding of quantum science. At the
same time the connection between quantum science and something as ubiquitous as
the television remote control is not immediately obvious. Thus, we have
combined hands-on experiences, visualizations, and traditional instruction to
help the students see these connections.
Students
should recognize these objects and see them in their everyday life. Light
emitting diodes (LEDs), for example, are everywhere. Although many students do
not know the name, they have seen them in their computers, remote controls, etc.
By examining the properties of LEDs the students learn that LEDs are different
from other light sources. Then, with the help of computer visualizations they
understand how the light emitting properties are related to the quantization of
energy in atoms. We occasionally use devices that students may have heard about
and may have seen pictures of, but they have probably not encountered. The
scanning tunnelling microscope is the best example. We do not expect students
to use a scanning tunnelling microscope although it is possible for students to
build one. But most students will not be able to build such a device. So, in
this case, we use a combination of a simulation and an interactive program
(Rebello et al 1997).
We also use
a variety of solid light sources. Infrared detector cards are a rather
interesting example. They are a fairly recent development – at least fairly
recent for inexpensive versions. TV repair people need to know if a television
remote control is emitting infrared. How can they do that? It is rather simple
if they have a video camera. The camera responds to IR and shows a bright spot
where the IR is emitted. So, every TV repairperson needs a video camera, and
he/she can find out whether there is light coming out of the remote control. But
that is rather expensive. Another way to detect IR is with rattlesnakes, which
are sensitive to infrared. So, every TV repairperson could have a rattlesnake.
But that is rather expensive in a different way. However, one can buy a little
card that responds to IR by emitting visible light. Thus, it absorbs low energy
light and emits higher energy light.
The Star
Trek Transporter is also a quantum mechanical device. If one reads the Star
Trek Users’ Manual, one finds that the Transporter has a component called a
Heisenberg Compensator (Sternbach 1991). When one of the writers for Star Trek
was asked, "How does the
Heisenberg Compensator work," he responded, "Very well" (Time
1994). Because Werner Heisenberg is one of the founders of quantum science, we
must assume that this Compensator is related to his Uncertainty Principle. In
one of our units we ask students to address the fantasy device in terms of
basic quantum mechanics principles. These and several other devices are
introduced to students. In each case we show how the devices are related to
quantum mechanics. Further, the students learn how the devices work at the
atomic level.
4. Using
visualization & model building
In the Visual
Quantum Mechanics instructional materials we provide as concrete a description
as possible about how we know about atoms and how we use that knowledge to
build models. One of our learning units focuses on spectroscopy and its role as
evidence for energy quantization. This unit begin with a study of the light
emitting diode (LED). We can convince students that the LED is related to
contemporary physics because they can read statements such as “A genuine White
Light Super Bright LED … utilizes an advanced Quantum Well technology…”
(Electronics 2000) After observing how different colours of LEDs respond to
changes in voltage and observing the spectra from both LEDs and gas spectral
tubes, the students are ready to build an energy level model of the atom. A
visualization program, Spectroscopy Lab Suite, provides a set of simulated
experiments, similar to the ones that they have just done, that are coupled to
building energy models of atoms. The activities include the emission and
absorption of light by gases, the emission by solids – particularly LEDs,
several types of lasers, and common emission processes such as fluorescence and
phosphorescence (Rebello et al 1998).
Students
generally use the Gas Emission program after observing the spectra
emitted by gas discharge tubes. The design of this component was motivated by
the results from a preliminary field test. We found that students related the
spectral lines for a gas to the discrete energy levels, rather than transitions
between these energy levels. The Emission module was created to alleviate this
misconception. Students create a trial spectrum for a gas by manipulating the
energy level diagram of a gas, indicating the transitions on it. They compare
their trial spectrum with the real spectrum for the gas.
The
component screen for the Emission module shows an array of simulated gas
lamps and a power supply on the left. To create the feel of the real experiment
that the students have already completed, they must drag one of the gas lamps
into the power supply. This action causes the lamp to emit light and its
spectrum appears at the top of the screen. There are five known gases
(hydrogen, helium, neon, lithium and mercury) available to the student, and an
unknown gas. In the case of the unknown gas, the student can change the
spectral lines to create any hypothetical spectrum.
A scale,
which represents energy in the atom, is displayed on the lower right side of
the screen. The students’ task is to manipulate energy levels and transitions
and reproduce the spectrum of the gas. This procedure addresses our research
about students’ understanding of atoms. Students can move the energy levels and
observe the corresponding changes in the energy of the spectral line. To make
the spectral line in the trial spectrum coincide with one in the real spectrum,
the students must create a transition between two energy levels whose
difference in energies is equal to the energy of the emitted light.
By using
this program students learn that the energy of the emitted light is equal to
the change in energy within the atom. More importantly they see that only
certain discrete energy levels are needed to explain the observed spectrum.
From knowledge of energy conservation and the data presented by the spectrum of
a gas, students can discover that energy states in atoms are quantized. This
critical discovery of 20thCentury physics follows from empirical results and an
explanation in terms of energy – no knowledge of wave functions or the Bohr
Atom is needed.
The Gas
Lamps Emission component does not enable students to determine the exact
energy levels of a given gas, but rather construct a model based on energy
differences. When this component is used in a classroom environment with
students working in small groups, different groups of students may arrive at
different energy levels within the models to explain the spectrum of the same
gas. Rather than tell some students that they are wrong, a teacher can use this
situation to discuss the nature of scientific models and limitations based on
the models by available data. In creating their models the students had
available to them only the observed spectrum and the conservation of energy.
With no further information they could create several different sets of energy
levels which match the data. By having students compare their results with
others in the class, they can begin to understand how more than one solution to
a problem can be “right” when it is based on limited information. However,
while they cannot create a complete picture of the atom, all students agree
that discrete energy levels are necessary.
Other modelling
in Spectroscopy Lab Suite is similarly connected to experiments. For
example, students create energy band and gap models for the observation that
LEDs which emit different colours of light have different threshold voltages.
They also interpret the behavior of electrons in conduction, valence and
impurity bands to explain why a glow-in-the-dark toothbrush stops glowing if it
is placed in liquid nitrogen. In all cases the energy model building is
connected directly to an observation that the students can make.
In building
the instruction that led to these programs, we expected the students to be
interacting with each other and with the teacher. For example, the observation
that different energy levels can give the same result is effective because two
groups of students obtain different but equally correct answers. The teacher
and the students’ peers are, thus, important to our teaching-learning process.
5.
Conceptual approaches to wave functions
Students
with concrete reasoning skills can move beyond spectra and energy models of the
atom to learning activities involving wave functions. Developing experiments
with real equipment to explore and apply wave functions is rather difficult.
However, we can create visualizations which help the students explore. For the
students who are not science or engineering majors we avoid the mathematics of
quantum mechanics and rely heavily on visualization in which the students
manipulate variables, and the computer solves Schrödinger’s Equation. The students
must then interpret the results in terms of the conceptual knowledge.
For both the
secondary students and beginning physics students we begin the study of the
wave nature of matter with an experimental observation – electrons can behave
as waves. After the students have discussed how interference patterns indicate
wave behavior and have observed the interference of light, we turn their
attention to electrons. They can observe a real experiment if the equipment is
available, use video simulations (Kirstein 1999) or see pictures in books. To
investigate the wave nature of electrons further, the students use a simulation
program which enables them to control variables in electron, two-slit
experiments. Using results such as those shown in Figure 3, the students can
discover a qualitative relation between the wavelength of the electron and its
energy. They compare the changes in the pattern for changes in energy of
electrons with similar changes when one observes the interference of light at
different wavelength. They can easily conclude that the wavelength of electrons
decreases as the energy increases.
An issue
that students will sometimes raise is the relation between the particle’s
charge and its wave behavior. Their reasoning is, “Diffraction is the spreading
out of a wave. Like charges repel. In a beam the charge must cause the
electrons to spread out.” We test this hypothesis by comparing the diffraction
pattern of simulated proton and neutron, two-slit experiments. The patterns are
identical, so charge must not be a factor.
After a few
more experiments, including a variation in mass, we introduce the de Broglie
equation.
We have not
actually derived the equation experimentally but have given a feasibility
argument for it. While this approach is not historically accurate, it seems to
provide students with a somewhat more concrete introduction to an abstract
concept than stating de Broglie’s hypothesis and then using interference
experiments to verify it.
To connect
the matter waves to probability we employ a simulated electron interference
experiment. Setting the particle flux to a few per second the students watch
the pattern develop. After a few particles have hit the screen, we have the
students stop the “experiment.” Now, we ask them to predict where the next
electron coming from our electron gun will appear on the screen. The students
very quickly fall into discussing the location in terms of probability. They
can indicate some location where the electron will rather definitely not appear
and several where it is very likely to appear. However, they cannot give a
definitive answer. Thus, we can introduce the wave function and its
probabilistic interpretation based on the students’ experience with
indeterminacy.
With wave
functions we emphasize conceptual understanding by having students manipulate
graphic images in accordance with their knowledge. For example, we ask the
students at all levels to sketch wave functions qualitatively. Following
procedures that appeared in French and Taylor (French & Taylor 1978), some
sketching is done with paper and pencil. However, we find that students can
easily be very inexact with paper and pencil, and sometimes exactness is
needed. So, we have created a program that does very little except that it allows
the students to vary the wave function and match boundary conditions.
We have
discovered some interesting ways in which the students use this program. First,
if we tell the students that the wave function is smooth, they will make it
smooth to many derivatives. The idea that two functions just stick together
does not occur to them. Second, we use of the word "decreasing" for
exponential decay. When we use "decay," the students immediately
think of radioactive decay. They interpret that to mean that the electrons are
radioactively decaying in the region where the total energy is less than the
potential energy. So, we use the phrase "decreasing wave function."
The third
and fourth year university physics students still use a basic Learning Cycle
style approach. However, the Explorations require a little bit of formal
operations. These students are still asked to match boundary conditions
graphically. However, the Wave Function Sketcher program for these
students also includes the common mathematical language of physicists.
In addition,
the students are expected to work with both the wave function and its
derivative when they are matching the boundary conditions. Using the Wave
Function Sketcher for the advanced students provides an intuitive, and
somewhat concrete, approach to understanding the process of matching boundary
conditions. After the students have completed these activities they are ready
to use the mathematics involved in boundary value problems to complete the
solutions for wave functions in various one-dimensional situations. While this
type of Learning Cycle primarily focuses on formal operations, it does provide
visualizations that are more concrete than typical mathematical symbols and, we
hope, helps the students build their intuition about wave functions.
6. Does
visual quantum mechanics work?
The units
have been used in secondary schools and in universities throughout the U.S. and
in a few other places. Actually, we do not know all the places that it is being
used because, during the field test phase, all the material was on the web and
people download it. We have given materials to people in Southeast Asia and
throughout various parts of Europe as well as the U.S. Most of the original
units have now been translated into Hebrew. (Arieli 2001) Thus, the materials,
except for the new Advanced Visual Quantum Mechanics, have been thoroughly
field-tested.
Most of our
reports, however, have come from the U.S. Approximately 175 different teachers
in 160 different schools have used the materials in classes and reported
results back to us. Students' attitudes toward these materials are very
positive. They frequently make comments like, "I really like this better
than our regular physics. Can we keep doing it?" (We don't tell the instructors
that.) Our staff hass observed teachers using the materials in a variety of
different schools. The students interact with the materials and each other; and
they seem to be learning. Most of the teachers also have positive attitudes; a
few do not. We certainly have the problem that many teachers in the U.S. do not
have a very strong background in quantum mechanics. Even though we are
approaching quantum mechanics in a much different way than it is normally
taught, some teachers still feel uncomfortable. Building the teachers'
confidence is very important. We are working on that aspect now by building a
Web-based course for secondary science teachers. (Connect to
http://kzollman1.phys.ksu.edu.)
Student
learning was also rather good. During our observations of the teaching, we
noticed that the hands-on component for both the real experiments and the
visualizations was important. Some teachers decided that it was too much
trouble to have the students work in a hands-on mode with all of these
programs. So, they just demonstrated the programs to the students. In these
cases learning went down; attitudes went down; everything went down. Hands-on
activities make a difference. Of course we should not be surprised because we
built the material for the students to use; not for the teacher to talk about.
With the
second year physics students we have done some testing, but it has not been as
extensive as for the secondary students. These units are somewhat shorter and
built to be in one to two-hour units within a traditional “modern physics”
course. We have found that the students’ attitudes are generally very positive
toward this type of learning and material. However, occasionally a student
would feel that he or she was not getting all of the material that he or she
would need for advanced level courses. Some of the first students with whom we
tested the material are now taking a fourth-year quantum mechanics course. We
will be investigating with them how well the materials that they learned in our
course are serving them in the more advanced course. In terms of the student
questions we found that these materials motivated students to ask very
high-level conceptual questions. Rather than most of the questions about wave
functions being concerned with the procedural efforts of manipulating
equations, the students were focused on what the wave function means and how it
can be interpreted. We asked the students questions on examinations which were
very similar to those that they might find in a higher-level course. In general
the performance on such questions was really quite good. So, overall even
though we have not tested the materials as carefully as the materials for lower
level students, we feel quite confident that the materiaals are teaching well
and are providing conceptual understanding through concrete hands-on and
visualized activities.
We do not at
this time have similar information about the materials for the fourth year
students. These materials are very new and have yet to have a significant
amount of classroom tests performed on them.
7.
Conclusions
Concerning
the questions which we posed in the title of this talk, we believe that the Visual
Quantum Mechanics project has shown that we can make quantum mechanics
accessible to students who are at the concrete operational stage. Further, we
believe that we must provide ways to allow students who are not formally
operational to begin to understand some of the features of the most important
scientific advances during the 20thcentury. Based on a large number of field
tests and a rather careful evaluation of student attitudes and learning, we
have concluded that the Visual Quantum Mechanics materials have been
successful in teaching some abstract concepts to students who have limited
science and mathematics background and who probably use concrete or
transitional operations. Our materials are also successful in teaching the
conceptual ideas of quantum mechanics to students who have stronger science and
engineering backgrounds by providing them with some concrete experiences. The
combination of hands-on activities, pencil and-paper exercises, and interactive
computer visualizations seem to work well in a classroom environment where
student-student and student-teacher interactions are taking place. Thus, we
feel that we have built a foundation for providing instruction in the most
important aspects of 20thCentury physics to a broad range of 21stCentury
students.
Acknowledgments
The work
described here has been supported primarily by the U.S. National Science
Foundation. Additional funding has come from the U.S. Department of Education,
the Eisenhower Professional Development Program and the Howard Hughes Medical
Institute. The development of the original Visual Quantum Mechanics
teaching materials profited from significant work by N. Sanjay Rebello,
Lawrence Escalada, and Michael Thoresen. Kirsten Hogg and Lei Bao were
instrumental in the development of materials for second year physics students,
while Waldemar Axmann is the primary author and programmer for Advanced Visual
Quantum Mechanics. Dr. Hogg has also been the primary author of the
Web-based materials while Kevin Zollman is the primary programmer for the
on-line course. Chandima Cumaranatunge programmed the visualizations described
in this paper. Rami Arieli has provided valuable feedback and suggestions for
improvement while he has been creating the Hebrew translation. We have worked
with Manfred Euler, IPN – Kiel, and Hartmut Wiesner, LMU – Munich, on some
aspects of Visual Quantum Mechanics. We have profited greatly from input
from undergraduate students and teachers at many other universities and high
schools where the Visual Quantum Mechanics materials have been tested.
References
Arieli R., Visual
Quantum Mechanics (Hebrew). Rehovot, Israel, Weizmann Institute of Science,
(2001).
Arons A., A
Guide to Introductory Physics Teaching, New York, John Wiley & Sons,
(1990).
Dick Smith,
Electronics, Flyer included with a white LED, Australia, (2000).
French A,
Taylor E., An Introduction to Quantum Physics, New York, W. W. Norton
& Co., (1978).
Karplus R.,
Science Teaching and the Development of Reasoning, Journal of Research in
Science Teaching, 14, (1977), 169.
Karplus R,
Renner J,
Fuller R, Collea F, Paldy L., Workshop on Physics Teaching and the Development
of Reasoning. Stony Brook: American Association of Physics Teachers, (1975).
Kirstein J.,
Interaktive Bildschirmexperimente, Ph.D. thesis, Technical University,
Berlin, (1999). McKinnon JW, Rennerr JW., Are colleges concerned about
intellectual development?, American Journal of Physics, 39, (1971), 1047-52.
Rebello NS,
Cumaranatunge C, Escalada L, Zollman D., Simulating the spectra of light
sources, Computers in Physics, 12, (1998), 28-33.
Rebello NS,
Sushenko K, Zollman D., Learning the physics of the scanning tunnelling
microscope using a computer program, European Journal of Physics, 18, (1997),
456-61.
Sternbach
R., Star Trek : The Next Generation Technical Manual, New York, Pocket
Books, (1991).
Time,
Reconfigure the Modulators! Time, (1994), 144.
Wells M,
Hestenes D, Swackhamer G., A Modeling Method for High School Physics
Instruction, American Journal of Physics, 63, (1995), 606-619.
Zollman D.,
Learning Cycles in a Large Enrollment Class, The Physics Teacher, 28, (1990),
20-5.
This paper
was presented to the First International GIREP Seminar, 2nd -
6th September, 2001, University of Udine, Italy.